This invention departs from the conventional way of solving the problem of zero-residual-energy minimum-time slew of a flexible space structure with damping using Modern Optimal Control tools. Instead of the prior art techniques presently in use, a minimum time, zero-residual-energy torque profile with unequal intervening pulses is arrived at heuristically. This is accomplished by a close examination of the response of an undamped and then damped structural mode to a sequence of step forces--entirely within the premises of structural dynamics discipline. Rigorous yet simple relationships are then established among the maneuver angle of a rest-to-rest slew, slew time, widths of the intervening pulses, and natural frequency and damping of a critical mode whose energy at the end of slew must be zero. Numerical results, illustrating the theory, demonstrate that a flexible space structure with small natural damping can be slewed with on-off thrusters such that a critical elastic mode has zero energy as the slew ends.
Over the last decade, significant strides have been made in the area of slewing flexible spacecraft. Several researchers have considered closed-loop linear optimal controllers with continuous torque profiles, with the objective of minimizing energy in a flexible mode at the end of slew. (See (1) Byers et al, Near Minimum Time, Closed-Loop Slewing Of Flexible Spacecraft, AIAA Journal of Guidance, Control, and Dynamic, Vol. 13, No. 1, January-February 1990, pp. 57-65, (2) Juang et al, A Slewing control Experiment for Flexible Structure, Journal of Guidance, Control, and Dynamics, Volume 9, September-October 1986, pp. 599-607, (3) Breakwell, Optimal Feedback Slewing of Flexible Spacecraft, Journal of Guidance and Control, Vol. 4, September-October 1981, pp. 472-479 and (4) Turner and Chun, Optimal Distributed Control of a Flexible Spacecraft During a Large Angle Maneuver", Journal of Guidance, Control and Dynamics, Vol. 7, May-June, 1984, pp. 157-264.) Meanwhile, other investigators have examined open-loop time-optimal, or nearly so, torque profiles to achieve the same objective. Two endeavors in this category are disclosed in papers by Singh et al, (See Planar, Time-Optimal Rest-to-Rest Slewing Maneuvers of Flexible Spacecraft, Journal of Guidance, Control and Dynamics, Vol. 12, No. 1, January-February, 1989, pp. 71-81), and Thompson et al, (See Near-Minimum-Time Open-Loop Slewing of Flexible Vehicles, Journal of Guidance, Control and Dynamics, Vol. 12, No. 1, January-February, 1989, pp. 82-88)
From the foregoing, and from other articles referenced therein, it has been observed that most investigators have formulated the problem within the framework of controls discipline: first-order state space model, application of Pontryagin's maximum principle, formation of Hamiltonian, Lagrange multipliers, costate vectors, and so forth. While this is a powerful approach and it yields elegant solutions, it overlooks the simple response of a second order structural mode to a bang-bang torque profile, and none of the prior investigators have proposed a zero-residual-energy minimum-time torque profile for slewing a flexible space structure with its small inherent damping.
A study of the response of an undamped structural mode excited by a sequence of step torques, reported below, shows a torque profile with compensating pulses such that it fulfills two requirements at once: (1) it slews a free space structure by a desired angle, and (2) it zeroes the energy in a critical flexible mode a the end of the slew. In addition, the relationships among the widths of intervening pulses, slew time, maneuver angle, critical modal frequency and structural damping are established and the numerical results illustrating the theory are also disclosed.